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proof that |g|g|g| divides expGexpG\operatorname{exp}~{}G

The following is a proof that, for every groupGGG that has an exponent and for every g∈GgGg\in G, |g|g|g|dividesexpGexpG\operatorname{exp}~{}G.

Proof.

By the division algorithm, there exist q,r∈ℤqrℤq,r\in{\mathbb{Z}} with 0≤r<|g|0rg0\leq r<|g| such that expG=q⁢|g|+rexpGqgr\operatorname{exp}~{}G=q|g|+r. Since eG=gexpG=gq⁢|g|+r=(g|g|)q⁢gr=(eG)q⁢gr=eG⁢gr=grsubscripteGsuperscriptgexpGsuperscriptgqgrsuperscriptsuperscriptggqsuperscriptgrsuperscriptsubscripteGqsuperscriptgrsubscripteGsuperscriptgrsuperscriptgre_{G}=g^{{\operatorname{exp}~{}G}}=g^{{q|g|+r}}=(g^{{|g|}})^{q}g^{r}=(e_{G})^{%
q}g^{r}=e_{G}g^{r}=g^{r}, by definition of the order of an element, rrr cannot be positive. Thus, r=0r0r=0. It follows that |g|g|g| divides expGexpG\operatorname{exp}~{}G.
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